Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Say I have some projective space $\mathbb{P}^n$ and some line bundle $L=\mathcal{O}(-k)$. Now, I want to have a subvariety $Y$ in $\mathbb{P}^n$ such that $L\vert_Y$ is trivial.

When is this the case? I can only think of trivial solutions, like when $Y$ is just a point and I can't seem to find a standard treatment of this in literature

share|improve this question

1 Answer 1

up vote 6 down vote accepted

Let's exclude the other trivial solution: $k=0$ and any $Y$.

Suppose now that $k\ne 0$. As $O(-k)|_Y$ trivial is equivalent to $O(k)|_Y$ (isomorphic to the dual of $O(-k)|_Y$) trivial, we can restrict to the case $k<0$. Then $L$, hence $L|_Y$, are ample. If $L|_Y$ is moreover trivial, then $O_Y$ is ample, which implies that $Y$ is affine. So necessarily $Y$ is affine. If $Y$ is a closed subvariety, this forces $Y$ to be a finite set.

Conclusion: if $Y$ is a closed subvariety such that $L|_Y$ is trivial with $k\ne 0$, then $Y$ is a finite subset.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.